The multivariate signed Bollobás–Riordan polynomial

We generalise the signed Bollobás–Riordan polynomial of S. Chmutov and I. Pak [S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás–Riordan polynomial, Mos. Math. J. 7(3) (2007), 409–418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial, J. Combin. Theory, Ser. B 99(3) (2009), 617–638. arXiv:0711.3490, doi:10.1016/j.jctb.2008.09.007] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.